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Eigenspaces of invariant differential operators on an affine symmetric space

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References

  1. Berger, M.: Les espaces symétriques non compacts. Ann. Sci. Ecole Norm. Sup.74, 85–177 (1957)

    Google Scholar 

  2. Bernstein, I.N., Gel'fand, S.I.: Meromorphic property of the functionP λ (in Russian). Funkt. Anal. i Ego Prilozheniya3, 84–85 (1969) English translation: Funct. Anal. Appl.,3, 68–69 (1969)

    Google Scholar 

  3. Bony, J.M.: Propagation des singularités différentiables pour des opérateurs à coefficient analytiques. Astérisque,34–35, 43–91 (1976)

    Google Scholar 

  4. Borel, A., de Siebenthal, J.: Les sous-groupes fermés de rang maximum des groupes de Lie clos. Comm. Math. Helv.,23, 200–221 (1949)

    Google Scholar 

  5. Bourbaki, N.: Groupes et Algèbres de Lie, Chapters IV–VI. Paris: Hermann 1968

    Google Scholar 

  6. Bruhat, F.: Sur les représentations induites des groupes de Lie. Bull. Soc. Math. France,84, 97–205 (1956)

    Google Scholar 

  7. Flensted-Jensen, M.: Spherical functions on a real semisimple Lie group. A method of reduction to the complex case. J. Functional Analysis,30, 106–146 (1978)

    Google Scholar 

  8. Furstenberg, H.: A Poisson formula for semisimple Lie groups. Ann. of Math.,77, 335–386 (1963)

    Google Scholar 

  9. Gindikin, S.G., Karpelevič, F.I.: Plancherel measure for Riemannian symmetric spaces of nonpositive curvature. Doklady Akad. Nauk. SSSR,145, 252–255 (1962). English translation: Soviet Math. Dokl.,3-II, 962–965 (1962)

    Google Scholar 

  10. Goto, M.: Faithful representation of Lie groups I. Math. Japonica,1, 107–119 (1948)

    Google Scholar 

  11. Harish-Chandra: Representations of semi-simple Lie groups I. Trans. Amer. Math. Soc.,75, 185–243 (1953)

    Google Scholar 

  12. Harish-Chandra: Spherical functions on a semi-simple Lie group I. Amer. J. Math.,80, 241–310 (1958)

    Google Scholar 

  13. Helgason, S.: Differential Geometry and Symmetric Spaces. New York: Academic Press 1962

    Google Scholar 

  14. Helgason, S.: A duality for symmetric spaces with applications to group representations. Advances in Math.,5, 1–154 (1970)

    Google Scholar 

  15. Helgason, S.: A duality for symmetric spaces with applications to group representations II, ibid,22, 187–219 (1976)

    Google Scholar 

  16. Kashiwara, M., Kawai, T.: Micro-hyperbolic pseudo-differential operators I. J. Math. Soc. Japan,27, 359–404 (1975)

    Google Scholar 

  17. Kashiwara, M., Kowata, A., Minemura, K., Okamoto, K., Oshima, T., Tanaka, M.: Eigenfunctions of invariant differential operators on a symmetric space. Ann. of Math.,107, 1–39 (1978)

    Google Scholar 

  18. Kashiwara, M., Oshima, T.: Systems of differential equations with regular singularities and their boundary value problems. Ann. of Math.,106, 145–200 (1977)

    Google Scholar 

  19. Kobayashi, S., Nomizu, K.: Foundations of Differential Geometry, vol. II, New York: Wiley-Interscience 1969

    Google Scholar 

  20. Koh, S.S.: On affine symmetric spaces. Trans. Amer. Math. Soc.,119 291–309 (1965)

    Google Scholar 

  21. Komatsu, H.: Projective and injective limits of weakly compact sequences of locally convex spaces. J. Math. Soc. Japan,19, 366–383 (1967)

    Google Scholar 

  22. Langlands, R.P.: On the functional equations satisfied by Eisenstein series. Lecture Notes in Math., No. 544, Berlin-Heidelberg-New York: Springer-Verlag 1976

    Google Scholar 

  23. Lewis, J.: Eigenfunctions on symmetric spaces with distribution-valued boundary forms, J. Functional Analysis29, 287–307 (1978)

    Google Scholar 

  24. Lojasiewicz, S.: Sur le problème de la division. Studia Mathematica, TomXVIII, 87–136 (1959)

    Google Scholar 

  25. Matsuki, T.: The orbits of affine symmetric spaces under the action of minimal parabolic subgroups. J. Math. Soc. Japan,31, 331–357 (1979)

    Google Scholar 

  26. Matsuki, T., Oshima, T.: Orbits on affine symmetric spaces under the action of the isotropy subgroups, to appear in J. Math. Soc. Japan

  27. Minemura, K., Oshima, T.: Boundary value problems with regular singularities and Helgason-Okamoto conjecture. Publ. RIMS, Kyoto Univ.,12 (Suppl.) 257–265 (1977)

    Google Scholar 

  28. Murakami, S.: Sur la classification des algèbres de Lie réelles et simples. Osaka J. Math.,2, 291–307 (1965)

    Google Scholar 

  29. Nomizu, K.: Invariant affine connections on homogeneous spaces. Amer. J. Math.76, 33–65 (1954)

    Google Scholar 

  30. Oshima, T.: A realization of Riemannian symmetric spaces. J. Math. Soc. Japan,30, 117–132 (1978)

    Google Scholar 

  31. Oshima, T., Sekiguchi, J.: Boundary value problem on symmetric homogeneous spaces. Proc. Japan Acad.,53 (Ser. A), 81–83 (1977)

    Google Scholar 

  32. Satake, I.: On representations and compactifications of symmetric Riemannian spaces. Ann. of Math.,71, 77–110 (1960)

    Google Scholar 

  33. Sato, F.: Eisenstein series for indefinite quadratic forms, preprint

  34. Sato, M.: Theory of hyperfunctions, I, II. J. Fac. Sci. Univ. of Tokyo, Sect. I,8, 139–193, 357–437 (1959)

    Google Scholar 

  35. Sato, M., Kawai, T., Kashiwara, M.: Micro-functions and pseudo-differential equations. Proc. Conf. at Katata, 1971. Lecture Notes in Math. No. 287, pp. 265–529. Berlin-Heidelberg-New York: Springer 1973

    Google Scholar 

  36. Schiffmann, G.: Intégrales d'entrelacement et fonctions de Whittaker. Bull. Soc. Math. France,99, 3–72 (1971)

    Google Scholar 

  37. Schwartz, L.: Théorie des distributions, I, II. Paris: Hermann 1950–1951

    Google Scholar 

  38. Sekiguchi, J.: Boundary value problem on hyperboloids to appear in Nagoya Math. J.

  39. Sekiguchi, J.: Invariant system of differential equations on a Siegel's upper half plane, preprint

  40. Selberg, A.: Discontinuous groups and harmonic analysis. Proc. Int. Congress of Math., Stockholm, pp. 177–189, 1962

  41. Wallach, N.: Harmonic Analysis on Homogeneous Spaces. New York: Marcel Dekker, Inc., 1973

    Google Scholar 

  42. Warner, G.: Harmonic Analysis on Semi-simple Lie groups, I, II. Berlin-Heidelberg-New York: Springer-Verlag 1972

    Google Scholar 

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Author notes

  1. Jiro Sekiguchi

    Present address: Department of Mathematics, Tokyo Metropolitan University, Fukazawa, Setagaya-ku, 158, Tokyo, Japan

Authors and Affiliations

  1. Institute for Advanced Study, 08540, Princeton, New Jersey

    Toshio Oshima & Jiro Sekiguchi

  2. Department of Mathematics College of General Education, University of Tokyo, Komaba, 153, Tokyo, Japan

    Toshio Oshima & Jiro Sekiguchi

  3. Department of Mathematics, Faculty of Science and Research Institute for Mathematical Sciences, Kyoto University, 606, Kyoto, Japan

    Toshio Oshima & Jiro Sekiguchi

Authors
  1. Toshio Oshima
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  2. Jiro Sekiguchi
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Supported in part by the National Science Foundation Grant.

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Oshima, T., Sekiguchi, J. Eigenspaces of invariant differential operators on an affine symmetric space. Invent Math 57, 1–81 (1980). https://doi.org/10.1007/BF01389818

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